Yes. Always wash your hands before picking up a baby and after diaper changes. You can use antibacterial soap or alcohol based gel. If you are using a gel, make sure your habds are dry before picking up the baby in order to avoid the baby inhaling alcohol fumes.
No. If a parent is going to hold his/her own baby, there is no need to sanitize the parent's hands (unless the parent is ill, such as having cold symptoms), because parents and babies in the same household already share the same "germs". Normal handwashing is a good habit, such as handwashing after playing with pets, after using the bathroom, after gardening, etc. . . .
Yes. Depends on what you did before picking up the baby, of course, but in general, it's a good idea.